Materials Science and Engineering introduction Chapter 15 Composites 7th ed

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Chapter
16
Composites
Top. ABS plastic having
a low glass transition temperature.
Used for containment and cosmetic
purposes.
Bidirectional layers. 45
fiberglass. Provide torsional
stiffness.
Unidirectional layers. 0 (and
some 90 ) fiberglass. Provide
longitudinal stiffness.
Side. ABS plastic
having a low glass
transition temperature.
Containment and
cosmetic.
Core wrap. Bidirectional
layer of fiberglass. Acts
as a torsion box and
bonds outer layers
to core.
Core. Polyurethane
plastic. Acts as a filler.
Bidirectional layer.
45 fiberglass.
Provides torsional
stiffness.
Damping layer. Polyurethane.
Improves chatter resistance.
Unidirectional layers. 0 (and
some 90 ) fiberglass. Provide
longitudinal stiffness.
Edge. Hardened
steel. Facilitates
turning by “cutting”
into the snow.
Bidirectional layer. 45 fiberglass.
Provides torsional stiffness.
Base. Compressed carbon
(carbon particles embedded
in a plastic matrix). Hard
and abrasion resistant. Provides
appropriate surface.
O
ne relatively complex composite structure is the modern ski. In this illustration, a cross section of a high-performance
snow ski, are shown the various components. The function of each component is noted, as well as the material that is used in
its construction. (Courtesy of Evolution Ski Company, Salt Lake City, Utah.)
WHY STUDY Composites?
With a knowledge of the various types of composites,
as well as an understanding of the dependence of
their behaviors on the characteristics, relative amounts,
geometry/distribution, and properties of the constituent phases, it is possible to design materials with
property combinations that are better than those
found in the metal alloys, ceramics, and polymeric
materials. For example, in Design Example 16.1, we
discuss how a tubular shaft is designed that meets
specified stiffness requirements.
• 577
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Learning Objectives
After careful study of this chapter you should be able to do the following:
1. Name the three main divisions of composite
5. Compute longitudinal strengths for discontinumaterials, and cite the distinguishing feature of
ous and aligned fibrous composite materials.
each.
6. Note the three common fiber reinforcements
2. Cite the difference in strengthening mechanism
used in polymer-matrix composites, and, for
for large-particle and dispersion-strengthened
each, cite both desirable characteristics and
particle-reinforced composites.
limitations.
3. Distinguish the three different types of fiber7. Cite the desirable features of metal-matrix
reinforced composites on the basis of fiber length
composites.
and orientation; comment on the distinctive me8. Note the primary reason for the creation of
chanical characteristics for each type.
ceramic-matrix composites.
4. Calculate longitudinal modulus and longitudinal
9. Name and briefly describe the two subclassificastrength for an aligned and continuous fibertions of structural composites.
reinforced composite.
16.1 INTRODUCTION
principle of
combined action
Many of our modern technologies require materials with unusual combinations of
properties that cannot be met by the conventional metal alloys, ceramics, and polymeric
materials. This is especially true for materials that are needed for aerospace, underwater, and transportation applications. For example, aircraft engineers are increasingly
searching for structural materials that have low densities, are strong, stiff, and abrasion
and impact resistant, and are not easily corroded. This is a rather formidable combination of characteristics. Frequently, strong materials are relatively dense; also, increasing
the strength or stiffness generally results in a decrease in impact strength.
Material property combinations and ranges have been, and are yet being, extended by the development of composite materials. Generally speaking, a composite
is considered to be any multiphase material that exhibits a significant proportion
of the properties of both constituent phases such that a better combination of properties is realized. According to this principle of combined action, better property
combinations are fashioned by the judicious combination of two or more distinct
materials. Property trade-offs are also made for many composites.
Composites of sorts have already been discussed; these include multiphase
metal alloys, ceramics, and polymers. For example, pearlitic steels (Section 9.19)
have a microstructure consisting of alternating layers of a ferrite and cementite
(Figure 9.27). The ferrite phase is soft and ductile, whereas cementite is hard and
very brittle. The combined mechanical characteristics of the pearlite (reasonably
high ductility and strength) are superior to those of either of the constituent phases.
There are also a number of composites that occur in nature. For example, wood
consists of strong and flexible cellulose fibers surrounded and held together by a
stiffer material called lignin. Also, bone is a composite of the strong yet soft protein collagen and the hard, brittle mineral apatite.
A composite, in the present context, is a multiphase material that is artificially
made, as opposed to one that occurs or forms naturally. In addition, the constituent
phases must be chemically dissimilar and separated by a distinct interface. Thus,
most metallic alloys and many ceramics do not fit this definition because their multiple phases are formed as a consequence of natural phenomena.
In designing composite materials, scientists and engineers have ingeniously
combined various metals, ceramics, and polymers to produce a new generation of
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16.1 Introduction • 579
Matrix
phase
Dispersed
phase
(a)
( b)
(c )
(d)
(e)
Figure 16.1 Schematic representations of the various geometrical and spatial
characteristics of particles of the dispersed phase that may influence the properties of
composites: (a) concentration, (b) size, (c) shape, (d) distribution, and (e) orientation.
(From Richard A. Flinn and Paul K. Trojan, Engineering Materials and Their Applications,
4th edition. Copyright © 1990 by John Wiley & Sons, Inc. Adapted by permission of John
Wiley & Sons, Inc.)
matrix phase
dispersed phase
Figure 16.2 A
classification scheme
for the various
composite types
discussed in this
chapter.
extraordinary materials. Most composites have been created to improve combinations of mechanical characteristics such as stiffness, toughness, and ambient and
high-temperature strength.
Many composite materials are composed of just two phases; one is termed the
matrix, which is continuous and surrounds the other phase, often called the dispersed
phase. The properties of composites are a function of the properties of the constituent phases, their relative amounts, and the geometry of the dispersed phase.
“Dispersed phase geometry” in this context means the shape of the particles and
the particle size, distribution, and orientation; these characteristics are represented
in Figure 16.1.
One simple scheme for the classification of composite materials is shown in Figure 16.2, which consists of three main divisions: particle-reinforced, fiber-reinforced,
Composites
Particle-reinforced
Largeparticle
Dispersionstrengthened
Fiber-reinforced
Continuous
(aligned)
Structural
Discontinuous
(short)
Aligned
Laminates
Randomly
oriented
Sandwich
panels
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580 • Chapter 16 / Composites
and structural composites; also, at least two subdivisions exist for each. The dispersed phase for particle-reinforced composites is equiaxed (i.e., particle dimensions are approximately the same in all directions); for fiber-reinforced composites,
the dispersed phase has the geometry of a fiber (i.e., a large length-to-diameter ratio).
Structural composites are combinations of composites and homogeneous materials.
The discussion of the remainder of this chapter will be organized according to this
classification scheme.
Pa r t i c l e - Re i n f o r c e d C o m p o s i t e s
large-particle
composite
dispersionstrengthened
composite
As noted in Figure 16.2, large-particle and dispersion-strengthened composites
are the two subclassifications of particle-reinforced composites. The distinction
between these is based upon reinforcement or strengthening mechanism. The term
“large” is used to indicate that particle–matrix interactions cannot be treated on
the atomic or molecular level; rather, continuum mechanics is used. For most of
these composites, the particulate phase is harder and stiffer than the matrix. These
reinforcing particles tend to restrain movement of the matrix phase in the vicinity
of each particle. In essence, the matrix transfers some of the applied stress to the
particles, which bear a fraction of the load. The degree of reinforcement or improvement of mechanical behavior depends on strong bonding at the matrix–particle
interface.
For dispersion-strengthened composites, particles are normally much smaller,
with diameters between 0.01 and 0.1 mm (10 and 100 nm). Particle–matrix interactions that lead to strengthening occur on the atomic or molecular level. The mechanism of strengthening is similar to that for precipitation hardening discussed in
Section 11.9. Whereas the matrix bears the major portion of an applied load, the
small dispersed particles hinder or impede the motion of dislocations. Thus, plastic
deformation is restricted such that yield and tensile strengths, as well as hardness,
improve.
16.2 LARGE–PARTICLE COMPOSITES
rule of mixtures
For a two-phase
composite, modulus
of elasticity upperbound expression
Some polymeric materials to which fillers have been added (Section 15.21) are really
large-particle composites. Again, the fillers modify or improve the properties of the
material and/or replace some of the polymer volume with a less expensive material—
the filler.
Another familiar large-particle composite is concrete, which is composed of cement (the matrix), and sand and gravel (the particulates). Concrete is the discussion
topic of a succeeding section.
Particles can have quite a variety of geometries, but they should be of approximately the same dimension in all directions (equiaxed). For effective reinforcement,
the particles should be small and evenly distributed throughout the matrix. Furthermore, the volume fraction of the two phases influences the behavior; mechanical
properties are enhanced with increasing particulate content. Two mathematical
expressions have been formulated for the dependence of the elastic modulus on the
volume fraction of the constituent phases for a two-phase composite. These rule of
mixtures equations predict that the elastic modulus should fall between an upper
bound represented by
Ec 1u2 ⫽ EmVm ⫹ EpVp
(16.1)
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16.2 Large–Particle Composites • 581
55
50
45
300
40
Upper bound
250
35
30
200
Lower bound
150
25
Modulus of elasticity (106 psi)
Modulus of elasticity (GPa)
350
20
0
20
40
60
80
15
100
Tungsten concentration (vol%)
Figure 16.3 Modulus of
elasticity versus volume
percent tungsten for a
composite of tungsten
particles dispersed within a
copper matrix. Upper and
lower bounds are according
to Equations 16.1 and 16.2;
experimental data points are
included. (From R. H. Krock,
ASTM Proceedings, Vol. 63,
1963. Copyright ASTM, 1916
Race Street, Philadelphia, PA
19103. Reprinted with
permission.)
and a lower bound, or limit,
For a two-phase
composite, modulus
of elasticity lowerbound expression
cermet
Ec 1l2 ⫽
EmEp
VmEp ⫹ VpEm
(16.2)
In these expressions, E and V denote the elastic modulus and volume fraction,
respectively, whereas the subscripts c, m, and p represent composite, matrix, and
particulate phases. Figure 16.3 plots upper- and lower-bound Ec-versus-Vp curves
for a copper–tungsten composite, in which tungsten is the particulate phase;
experimental data points fall between the two curves. Equations analogous to 16.1
and 16.2 for fiber-reinforced composites are derived in Section 16.5.
Large-particle composites are utilized with all three material types (metals, polymers, and ceramics). The cermets are examples of ceramic–metal composites. The
most common cermet is the cemented carbide, which is composed of extremely hard
particles of a refractory carbide ceramic such as tungsten carbide (WC) or titanium
carbide (TiC), embedded in a matrix of a metal such as cobalt or nickel. These composites are utilized extensively as cutting tools for hardened steels. The hard carbide
particles provide the cutting surface but, being extremely brittle, are not themselves
capable of withstanding the cutting stresses.Toughness is enhanced by their inclusion
in the ductile metal matrix, which isolates the carbide particles from one another
and prevents particle-to-particle crack propagation. Both matrix and particulate
phases are quite refractory, to withstand the high temperatures generated by the
cutting action on materials that are extremely hard. No single material could possibly provide the combination of properties possessed by a cermet. Relatively large
volume fractions of the particulate phase may be utilized, often exceeding 90 vol%;
thus the abrasive action of the composite is maximized. A photomicrograph of a
WC–Co cemented carbide is shown in Figure 16.4.
Both elastomers and plastics are frequently reinforced with various particulate
materials. Our use of many of the modern rubbers would be severely restricted without reinforcing particulate materials such as carbon black. Carbon black consists of
very small and essentially spherical particles of carbon, produced by the combustion
of natural gas or oil in an atmosphere that has only a limited air supply. When added
to vulcanized rubber, this extremely inexpensive material enhances tensile strength,
toughness, and tear and abrasion resistance. Automobile tires contain on the order
of 15 to 30 vol% of carbon black. For the carbon black to provide significant
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Figure 16.4 Photomicrograph of a WC–Co
cemented carbide. Light areas are the
cobalt matrix; dark regions, the particles
of tungsten carbide. 100⫻. (Courtesy of
Carboloy Systems Department, General
Electric Company.)
reinforcement, the particle size must be extremely small, with diameters between
20 and 50 nm; also, the particles must be evenly distributed throughout the rubber
and must form a strong adhesive bond with the rubber matrix. Particle reinforcement using other materials (e.g., silica) is much less effective because this special
interaction between the rubber molecules and particle surfaces does not exist. Figure 16.5 is an electron micrograph of a carbon black-reinforced rubber.
Concrete
concrete
Concrete is a common large-particle composite in which both matrix and dispersed
phases are ceramic materials. Since the terms “concrete” and “cement” are sometimes
incorrectly interchanged, perhaps it is appropriate to make a distinction between them.
In a broad sense, concrete implies a composite material consisting of an aggregate of
particles that are bound together in a solid body by some type of binding medium,
that is, a cement. The two most familiar concretes are those made with portland and
Figure 16.5 Electron micrograph showing
the spherical reinforcing carbon black
particles in a synthetic rubber tire tread
compound. The areas resembling water
marks are tiny air pockets in the rubber.
80,000⫻. (Courtesy of Goodyear Tire &
Rubber Company.)
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16.2 Large–Particle Composites • 583
asphaltic cements, where the aggregate is gravel and sand. Asphaltic concrete is widely
used primarily as a paving material, whereas portland cement concrete is employed
extensively as a structural building material. Only the latter is treated in this discussion.
Portland Cement Concrete
The ingredients for this concrete are portland cement, a fine aggregate (sand), a
coarse aggregate (gravel), and water. The process by which portland cement is produced and the mechanism of setting and hardening were discussed very briefly in
Section 13.7. The aggregate particles act as a filler material to reduce the overall
cost of the concrete product because they are cheap, whereas cement is relatively
expensive. To achieve the optimum strength and workability of a concrete mixture,
the ingredients must be added in the correct proportions. Dense packing of the aggregate and good interfacial contact are achieved by having particles of two different sizes; the fine particles of sand should fill the void spaces between the gravel
particles. Ordinarily these aggregates comprise between 60% and 80% of the total
volume. The amount of cement–water paste should be sufficient to coat all the sand
and gravel particles, otherwise the cementitious bond will be incomplete. Furthermore, all the constituents should be thoroughly mixed. Complete bonding between
cement and the aggregate particles is contingent upon the addition of the correct
quantity of water. Too little water leads to incomplete bonding, and too much results in excessive porosity; in either case the final strength is less than the optimum.
The character of the aggregate particles is an important consideration. In particular, the size distribution of the aggregates influences the amount of cement–water
paste required. Also, the surfaces should be clean and free from clay and silt, which
prevent the formation of a sound bond at the particle surface.
Portland cement concrete is a major material of construction, primarily because
it can be poured in place and hardens at room temperature, and even when submerged in water. However, as a structural material, there are some limitations and
disadvantages. Like most ceramics, portland cement concrete is relatively weak and
extremely brittle; its tensile strength is approximately 10 to 15 times smaller than
its compressive strength. Also, large concrete structures can experience considerable thermal expansion and contraction with temperature fluctuations. In addition,
water penetrates into external pores, which can cause severe cracking in cold
weather as a consequence of freeze–thaw cycles. Most of these inadequacies may
be eliminated or at least improved by reinforcement and/or the incorporation of
additives.
Reinforced Concrete
The strength of portland cement concrete may be increased by additional reinforcement. This is usually accomplished by means of steel rods, wires, bars (rebar),
or mesh, which are embedded into the fresh and uncured concrete. Thus, the reinforcement renders the hardened structure capable of supporting greater tensile,
compressive, and shear stresses. Even if cracks develop in the concrete, considerable reinforcement is maintained.
Steel serves as a suitable reinforcement material because its coefficient of thermal expansion is nearly the same as that of concrete. In addition, steel is not rapidly
corroded in the cement environment, and a relatively strong adhesive bond is
formed between it and the cured concrete. This adhesion may be enhanced by the
incorporation of contours into the surface of the steel member, which permits a
greater degree of mechanical interlocking.
Portland cement concrete may also be reinforced by mixing into the fresh concrete fibers of a high-modulus material such as glass, steel, nylon, and polyethylene.
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prestressed concrete
Care must be exercised in utilizing this type of reinforcement, since some fiber materials experience rapid deterioration when exposed to the cement environment.
Still another reinforcement technique for strengthening concrete involves the
introduction of residual compressive stresses into the structural member; the resulting material is called prestressed concrete. This method utilizes one characteristic
of brittle ceramics—namely, that they are stronger in compression than in tension.
Thus, to fracture a prestressed concrete member, the magnitude of the precompressive stress must be exceeded by an applied tensile stress.
In one such prestressing technique, high-strength steel wires are positioned inside
the empty molds and stretched with a high tensile force, which is maintained constant.
After the concrete has been placed and allowed to harden, the tension is released.
As the wires contract, they put the structure in a state of compression because the
stress is transmitted to the concrete via the concrete–wire bond that is formed.
Another technique is also utilized in which stresses are applied after the concrete hardens; it is appropriately called posttensioning. Sheet metal or rubber tubes
are situated inside and pass through the concrete forms, around which the concrete
is cast. After the cement has hardened, steel wires are fed through the resulting
holes, and tension is applied to the wires by means of jacks attached and abutted
to the faces of the structure. Again, a compressive stress is imposed on the concrete
piece, this time by the jacks. Finally, the empty spaces inside the tubing are filled
with a grout to protect the wire from corrosion.
Concrete that is prestressed should be of a high quality, with a low shrinkage
and a low creep rate. Prestressed concretes, usually prefabricated, are commonly
used for highway and railway bridges.
16.3 DISPERSION-STRENGTHENED COMPOSITES
Metals and metal alloys may be strengthened and hardened by the uniform dispersion of several volume percent of fine particles of a very hard and inert material. The
dispersed phase may be metallic or nonmetallic; oxide materials are often used.Again,
the strengthening mechanism involves interactions between the particles and dislocations within the matrix, as with precipitation hardening. The dispersion strengthening effect is not as pronounced as with precipitation hardening; however, the
strengthening is retained at elevated temperatures and for extended time periods because the dispersed particles are chosen to be unreactive with the matrix phase. For
precipitation-hardened alloys, the increase in strength may disappear upon heat treatment as a consequence of precipitate growth or dissolution of the precipitate phase.
The high-temperature strength of nickel alloys may be enhanced significantly
by the addition of about 3 vol% of thoria (ThO2) as finely dispersed particles; this
material is known as thoria-dispersed (or TD) nickel. The same effect is produced
in the aluminum–aluminum oxide system. A very thin and adherent alumina coating
is caused to form on the surface of extremely small (0.1 to 0.2 mm thick) flakes of
aluminum, which are dispersed within an aluminum metal matrix; this material is
termed sintered aluminum powder (SAP).
Concept Check 16.1
Cite the general difference in strengthening mechanism between large-particle and
dispersion-strengthened particle-reinforced composites.
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]
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16.4 Influence of Fiber Length • 585
Fi b e r- Re i n f o r c e d C o m p o s i t e s
fiber-reinforced
composite
specific strength
specific modulus
Technologically, the most important composites are those in which the dispersed
phase is in the form of a fiber. Design goals of fiber-reinforced composites often
include high strength and/or stiffness on a weight basis. These characteristics are expressed in terms of specific strength and specific modulus parameters, which correspond, respectively, to the ratios of tensile strength to specific gravity and modulus
of elasticity to specific gravity. Fiber-reinforced composites with exceptionally high
specific strengths and moduli have been produced that utilize low-density fiber and
matrix materials.
As noted in Figure 16.2, fiber-reinforced composites are subclassified by fiber
length. For short fiber, the fibers are too short to produce a significant improvement
in strength.
16.4 INFLUENCE OF FIBER LENGTH
The mechanical characteristics of a fiber-reinforced composite depend not only on
the properties of the fiber, but also on the degree to which an applied load is transmitted to the fibers by the matrix phase. Important to the extent of this load transmittance is the magnitude of the interfacial bond between the fiber and matrix
phases. Under an applied stress, this fiber–matrix bond ceases at the fiber ends,
yielding a matrix deformation pattern as shown schematically in Figure 16.6; in other
words, there is no load transmittance from the matrix at each fiber extremity.
Some critical fiber length is necessary for effective strengthening and stiffening
of the composite material. This critical length lc is dependent on the fiber diameter
d and its ultimate (or tensile) strength s*f , and on the fiber–matrix bond strength
(or the shear yield strength of the matrix, whichever is smaller) tc according to
Critical fiber
length—dependence
on fiber strength
and diameter, and
fiber-matrix bond
strength/matrix shear
yield strength
lc ⫽
s*f d
(16.3)
2tc
For a number of glass and carbon fiber–matrix combinations, this critical length is
on the order of 1 mm, which ranges between 20 and 150 times the fiber diameter.
When a stress equal to s*f is applied to a fiber having just this critical length,
the stress–position profile shown in Figure 16.7a results; that is, the maximum fiber
load is achieved only at the axial center of the fiber. As fiber length l increases, the
fiber reinforcement becomes more effective; this is demonstrated in Figure 16.7b,
a stress–axial position profile for l 7 lc when the applied stress is equal to the fiber
strength. Figure 16.7c shows the stress–position profile for l 6 lc.
Fibers for which l W lc (normally l 7 15lc) are termed continuous; discontinuous or short fibers have lengths shorter than this. For discontinuous fibers of lengths
␴
Matrix
␴
Fiber
␴
Figure 16.6 The
deformation pattern in
the matrix surrounding
a fiber that is subjected
to an applied tensile
load.
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586 • Chapter 16 / Composites
Maximum
applied load
␴*f
Stress
␴f*
Stress
Figure 16.7
Stress–position
profiles when fiber
length l (a) is equal
to the critical length
lc, (b) is greater than
the critical length,
and (c) is less than
the critical length for
a fiber-reinforced
composite that is
subjected to a tensile
stress equal to the
fiber tensile
strength s*f .
0
lc
lc
lc
2
2
2
Position
0
l
␴f*
␴f*
lc
2
Position
␴f*
l
␴f*
l = lc
l > lc
(a)
(b)
Stress
␴*f
0
Position
␴f*
l
␴f*
l < lc
(c)
significantly less than lc, the matrix deforms around the fiber such that there is virtually no stress transference and little reinforcement by the fiber. These are essentially
the particulate composites as described above. To affect a significant improvement
in strength of the composite, the fibers must be continuous.
16.5 INFLUENCE OF FIBER ORIENTATION
AND CONCENTRATION
The arrangement or orientation of the fibers relative to one another, the fiber concentration, and the distribution all have a significant influence on the strength and
other properties of fiber-reinforced composites. With respect to orientation, two
extremes are possible: (1) a parallel alignment of the longitudinal axis of the fibers
in a single direction, and (2) a totally random alignment. Continuous fibers are normally aligned (Figure 16.8a), whereas discontinuous fibers may be aligned (Figure
16.8b), randomly oriented (Figure 16.8c), or partially oriented. Better overall composite properties are realized when the fiber distribution is uniform.
Continuous and Aligned Fiber Composites
Tensile Stress–Strain Behavior—Longitudinal Loading
Mechanical responses of this type of composite depend on several factors to
include the stress–strain behaviors of fiber and matrix phases, the phase volume
fractions, and, in addition, the direction in which the stress or load is applied.
Furthermore, the properties of a composite having its fibers aligned are highly
anisotropic, that is, dependent on the direction in which they are measured. Let
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16.5 Influence of Fiber Orientation and Concentration • 587
Figure 16.8 Schematic
representations of
(a) continuous and
aligned, (b) discontinuous
and aligned, and
(c) discontinuous and
randomly oriented fiberreinforced composites.
Longitudinal
direction
Transverse
direction
(a)
(c)
us first consider the stress–strain behavior for the situation wherein the stress is
applied along the direction of alignment, the longitudinal direction, which is
indicated in Figure 16.8a.
To begin, assume the stress versus strain behaviors for fiber and matrix phases
that are represented schematically in Figure 16.9a; in this treatment we consider the
fiber to be totally brittle and the matrix phase to be reasonably ductile. Also indicated
␴*f
Fiber
Fiber
Stage
I
Ef
Composite
*
␴cl
*
␴m
Failure
Matrix
Stress
Stress
longitudinal
direction
(b)
Matrix
'
␴m
Stage
II
Em
⑀*f
Strain
(a)
*
⑀m
⑀ ym
⑀*f
Strain
(b)
Figure 16.9 (a) Schematic stress–strain curves for brittle fiber and ductile matrix
materials. Fracture stresses and strains for both materials are noted. (b) Schematic
stress–strain curve for an aligned fiber-reinforced composite that is exposed to a uniaxial
stress applied in the direction of alignment; curves for the fiber and matrix materials
shown in part (a) are also superimposed.
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588 • Chapter 16 / Composites
in this figure are fracture strengths in tension for fiber and matrix, s*f and sm
*,
respectively, and their corresponding fracture strains, ⑀*f and ⑀*;
furthermore,
it
is
m
assumed that ⑀m
which
is
normally
the
case.
* 7 ⑀*,
f
A fiber-reinforced composite consisting of these fiber and matrix materials will
exhibit the uniaxial stress–strain response illustrated in Figure 16.9b; the fiber and
matrix behaviors from Figure 16.9a are included to provide perspective. In the initial Stage I region, both fibers and matrix deform elastically; normally this portion
of the curve is linear. Typically, for a composite of this type, the matrix yields and
deforms plastically (at ⑀ym, Figure 16.9b) while the fibers continue to stretch elastically, inasmuch as the tensile strength of the fibers is significantly higher than the
yield strength of the matrix. This process constitutes Stage II as noted in the figure;
this stage is ordinarily very nearly linear, but of diminished slope relative to Stage
I. Furthermore, in passing from Stage I to Stage II, the proportion of the applied
load that is borne by the fibers increases.
The onset of composite failure begins as the fibers start to fracture, which
corresponds to a strain of approximately ⑀*f as noted in Figure 16.9b. Composite
failure is not catastrophic for a couple of reasons. First, not all fibers fracture at the
same time, since there will always be considerable variations in the fracture strength
of brittle fiber materials (Section 12.8). In addition, even after fiber failure, the matrix is still intact inasmuch as ⑀*f 6 ⑀m
* (Figure 16.9a). Thus, these fractured fibers,
which are shorter than the original ones, are still embedded within the intact matrix, and consequently are capable of sustaining a diminished load as the matrix
continues to plastically deform.
Elastic Behavior—Longitudinal Loading
Let us now consider the elastic behavior of a continuous and oriented fibrous composite that is loaded in the direction of fiber alignment. First, it is assumed that the
fiber–matrix interfacial bond is very good, such that deformation of both matrix
and fibers is the same (an isostrain situation). Under these conditions, the total load
sustained by the composite Fc is equal to the sum of the loads carried by the matrix phase Fm and the fiber phase Ff, or
Fc ⫽ Fm ⫹ Ff
(16.4)
From the definition of stress, Equation 6.1, F ⫽ sA; and thus expressions for Fc,
Fm, and Ff in terms of their respective stresses (sc, sm, and sf ) and cross-sectional
areas ( Ac, Am, and Af ) are possible. Substitution of these into Equation 16.4 yields
sc Ac ⫽ smAm ⫹ sf Af
(16.5)
and then, dividing through by the total cross-sectional area of the composite, Ac,
we have
Af
Am
⫹ sf
sc ⫽ sm
(16.6)
Ac
Ac
where AmⲐAc and Af ⲐAc are the area fractions of the matrix and fiber phases, respectively. If the composite, matrix, and fiber phase lengths are all equal, AmⲐAc is
equivalent to the volume fraction of the matrix, Vm, and likewise for the fibers,
Vf ⫽ AfⲐAc. Equation 16.6 now becomes
sc ⫽ smVm ⫹ sfVf
(16.7)
The previous assumption of an isostrain state means that
⑀c ⫽ ⑀m ⫽ ⑀f
(16.8)
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16.5 Influence of Fiber Orientation and Concentration • 589
and when each term in Equation 16.7 is divided by its respective strain,
sf
sm
sc
⫽
Vm ⫹
V
⑀c
⑀m
⑀f f
(16.9)
Furthermore, if composite, matrix, and fiber deformations are all elastic, then
sc Ⲑ⑀c ⫽ Ec, sm Ⲑ⑀m ⫽ Em, and sf Ⲑ⑀f ⫽ Ef, the E’s being the moduli of elasticity for
the respective phases. Substitution into Equation 16.9 yields an expression for the
modulus of elasticity of a continuous and aligned fibrous composite in the direction
of alignment (or longitudinal direction), Ecl, as
For a continuous
and aligned
fiber-reinforced
composite, modulus
of elasticity in the
longitudinal
direction
or
Ecl ⫽ EmVm ⫹ EfVf
(16.10a)
Ecl ⫽ Em 11 ⫺ Vf 2 ⫹ EfVf
(16.10b)
since the composite consists of only matrix and fiber phases; that is, Vm ⫹ Vf ⫽ 1.
Thus, Ecl is equal to the volume-fraction weighted average of the moduli of
elasticity of the fiber and matrix phases. Other properties, including density, also
have this dependence on volume fractions. Equation 16.10a is the fiber analogue of
Equation 16.1, the upper bound for particle-reinforced composites.
It can also be shown, for longitudinal loading, that the ratio of the load carried
by the fibers to that carried by the matrix is
Ratio of load carried
by fibers and the
matrix phase, for
longitudinal loading
Ff
Fm
⫽
EfVf
EmVm
(16.11)
The demonstration is left as a homework problem.
EXAMPLE PROBLEM 16.1
Property Determinations for a Glass Fiber-Reinforced
Composite—Longitudinal Direction
A continuous and aligned glass fiber-reinforced composite consists of 40 vol%
of glass fibers having a modulus of elasticity of 69 GPa (10 ⫻ 106 psi) and
60 vol% of a polyester resin that, when hardened, displays a modulus of
3.4 GPa (0.5 ⫻ 106 psi).
(a) Compute the modulus of elasticity of this composite in the longitudinal
direction.
(b) If the cross-sectional area is 250 mm2 (0.4 in.2) and a stress of 50 MPa
(7250 psi) is applied in this longitudinal direction, compute the magnitude of
the load carried by each of the fiber and matrix phases.
(c) Determine the strain that is sustained by each phase when the stress in
part (b) is applied.
Solution
(a) The modulus of elasticity of the composite is calculated using Equation
16.10a:
Ecl ⫽ 13.4 GPa210.62 ⫹ 169 GPa2 10.42
⫽ 30 GPa 14.3 ⫻ 106 psi2
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(b) To solve this portion of the problem, first find the ratio of fiber load to
matrix load, using Equation 16.11; thus,
Ff
Fm
⫽
169 GPa2 10.42
⫽ 13.5
13.4 GPa2 10.62
or Ff ⫽ 13.5 Fm.
In addition, the total force sustained by the composite Fc may be computed
from the applied stress s and total composite cross-sectional area Ac according to
Fc ⫽ Acs ⫽ 1250 mm2 2150 MPa2 ⫽ 12,500 N 12900 lbf 2
However, this total load is just the sum of the loads carried by fiber and matrix
phases; that is,
Fc ⫽ Ff ⫹ Fm ⫽ 12,500 N 12900 lbf 2
Substitution for Ff from the above yields
13.5 Fm ⫹ Fm ⫽ 12,500 N
or
whereas
Fm ⫽ 860 N 1200 lbf 2
Ff ⫽ Fc ⫺ Fm ⫽ 12,500 N ⫺ 860 N ⫽ 11,640 N 12700 lbf 2
Thus, the fiber phase supports the vast majority of the applied load.
(c) The stress for both fiber and matrix phases must first be calculated. Then,
by using the elastic modulus for each (from part a), the strain values may be
determined.
For stress calculations, phase cross-sectional areas are necessary:
and
Thus,
Am ⫽ Vm Ac ⫽ 10.62 1250 mm2 2 ⫽ 150 mm2 10.24 in.2 2
Af ⫽ Vf Ac ⫽ 10.42 1250 mm2 2 ⫽ 100 mm2 10.16 in.2 2
Fm
860 N
⫽ 5.73 MPa 1833 psi2
⫽
Am
150 mm2
Ff
11,640 N
⫽ 116.4 MPa 116,875 psi2
⫽
sf ⫽
Af
100 mm2
sm ⫽
Finally, strains are computed as
sm
5.73 MPa
⫽ 1.69 ⫻ 10⫺3
⫽
Em
3.4 ⫻ 103 MPa
sf
116.4 MPa
⫽ 1.69 ⫻ 10⫺3
⫽
⑀f ⫽
Ef
69 ⫻ 103 MPa
⑀m ⫽
Therefore, strains for both matrix and fiber phases are identical, which
they should be, according to Equation 16.8 in the previous development.
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16.5 Influence of Fiber Orientation and Concentration • 591
Elastic Behavior—Transverse Loading
transverse direction
A continuous and oriented fiber composite may be loaded in the transverse direction; that is, the load is applied at a 90⬚ angle to the direction of fiber alignment as
shown in Figure 16.8a. For this situation the stress s to which the composite as well
as both phases are exposed is the same, or
sc ⫽ sm ⫽ sf ⫽ s
(16.12)
This is termed an isostress state. Also, the strain or deformation of the entire composite ⑀c is
⑀c ⫽ ⑀mVm ⫹ ⑀fVf
(16.13)
s
s
s
⫽
Vm ⫹
V
Ect
Em
Ef f
(16.14)
but, since ⑀ ⫽ sⲐE,
where Ect is the modulus of elasticity in the transverse direction. Now, dividing
through by s yields
For a continuous
and aligned
fiber-reinforced
composite, modulus
of elasticity in the
transverse direction
Vf
Vm
1
⫽
⫹
Ect
Em
Ef
(16.15)
which reduces to
Ect ⫽
EmEf
VmEf ⫹ Vf Em
⫽
EmEf
11 ⫺ Vf 2Ef ⫹ Vf Em
(16.16)
Equation 16.16 is analogous to the lower-bound expression for particulate composites, Equation 16.2.
EXAMPLE PROBLEM 16.2
Elastic Modulus Determination for a Glass Fiber-Reinforced
Composite—Transverse Direction
Compute the elastic modulus of the composite material described in Example
Problem 16.1, but assume that the stress is applied perpendicular to the
direction of fiber alignment.
Solution
According to Equation 16.16,
Ect ⫽
13.4 GPa2 169 GPa2
10.62169 GPa2 ⫹ 10.42 13.4 GPa2
⫽ 5.5 GPa 10.81 ⫻ 106 psi2
This value for Ect is slightly greater than that of the matrix phase but, from
Example Problem 16.1a, only approximately one-fifth of the modulus of elasticity along the fiber direction (Ecl), which indicates the degree of anisotropy
of continuous and oriented fiber composites.
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592 • Chapter 16 / Composites
Table 16.1 Typical Longitudinal and Transverse Tensile Strengths
for Three Unidirectional Fiber-Reinforced Composites.
The Fiber Content for Each Is Approximately 50 Vol%
Material
Glass–polyester
Carbon (high modulus)–epoxy
Kevlar™–epoxy
Longitudinal
Tensile
Strength (MPa)
Transverse
Tensile
Strength (MPa)
700
1000
1200
20
35
20
Source: D. Hull and T. W. Clyne, An Introduction to Composite Materials, 2nd
edition, Cambridge University Press, 1996, p. 179.
Longitudinal Tensile Strength
We now consider the strength characteristics of continuous and aligned fiberreinforced composites that are loaded in the longitudinal direction. Under these
circumstances, strength is normally taken as the maximum stress on the stress–strain
curve, Figure 16.9b; often this point corresponds to fiber fracture, and marks the
onset of composite failure. Table 16.1 lists typical longitudinal tensile strength values
for three common fibrous composites. Failure of this type of composite material is
a relatively complex process, and several different failure modes are possible. The
mode that operates for a specific composite will depend on fiber and matrix properties, and the nature and strength of the fiber–matrix interfacial bond.
If we assume that ⑀f* 6 ⑀m
* (Figure 16.9a), which is the usual case, then fibers
will fail before the matrix. Once the fibers have fractured, the majority of the load
that was borne by the fibers is now transferred to the matrix. This being the case,
it is possible to adapt the expression for the stress on this type of composite, Equation 16.7, into the following expression for the longitudinal strength of the composite, s*cl:
For a continuous
and aligned
fiber-reinforced
composite,
longitudinal strength
in tension
s*cl ⫽ s¿m 11 ⫺ Vf 2 ⫹ s*f Vf
(16.17)
Here s¿m is the stress in the matrix at fiber failure (as illustrated in Figure 16.9a)
and, as previously, sf* is the fiber tensile strength.
Transverse Tensile Strength
The strengths of continuous and unidirectional fibrous composites are highly
anisotropic, and such composites are normally designed to be loaded along the highstrength, longitudinal direction. However, during in-service applications transverse
tensile loads may also be present. Under these circumstances, premature failure may
result inasmuch as transverse strength is usually extremely low—it sometimes lies
below the tensile strength of the matrix. Thus, in actual fact, the reinforcing effect
of the fibers is a negative one. Typical transverse tensile strengths for three unidirectional composites are contained in Table 16.1.
Whereas longitudinal strength is dominated by fiber strength, a variety of factors will have a significant influence on the transverse strength; these factors include properties of both the fiber and matrix, the fiber–matrix bond strength, and
the presence of voids. Measures that have been employed to improve the transverse strength of these composites usually involve modifying properties of the
matrix.
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16.5 Influence of Fiber Orientation and Concentration • 593
Concept Check 16.2
In the table below are listed four hypothetical aligned fiber-reinforced composites
(labeled A through D), along with their characteristics. On the basis of these data,
rank the four composites from highest to lowest strength in the longitudinal direction, and then justify your ranking.
Composite
Fiber
Type
Vol. Fraction
Fibers
A
B
C
D
glass
glass
carbon
carbon
0.20
0.35
0.40
0.30
Fiber
Strength
(MPa)
3.5
3.5
5.5
5.5
⫻
⫻
⫻
⫻
103
103
103
103
Ave. Fiber
Length
(mm)
Critical
Length
(mm)
8
12
8
8
0.70
0.75
0.40
0.50
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]
Discontinuous and Aligned Fiber Composites
For a discontinuous
(l ⬎ lc) and aligned
fiber-reinforced
composite,
longitudinal strength
in tension
For a discontinuous
(l ⬍ lc) and aligned
fiber-reinforced
composite,
longitudinal strength
in tension
Even though reinforcement efficiency is lower for discontinuous than for continuous fibers, discontinuous and aligned fiber composites (Figure 16.8b) are becoming
increasingly more important in the commercial market. Chopped glass fibers are
used most extensively; however, carbon and aramid discontinuous fibers are also
employed. These short fiber composites can be produced having moduli of elasticity
and tensile strengths that approach 90% and 50%, respectively, of their continuous
fiber counterparts.
For a discontinuous and aligned fiber composite having a uniform distribution of fibers and in which l 7 lc, the longitudinal strength (s*cd) is given by the
relationship
s*cd ⫽ s*f Vf a1 ⫺
lc
b ⫹ s¿m 11 ⫺ Vf 2
2l
(16.18)
where sf* and s¿m represent, respectively, the fracture strength of the fiber and the
stress in the matrix when the composite fails (Figure 16.9a).
If the fiber length is less than critical (l 6 lc), then the longitudinal strength
(s*
cd¿) is given by
s*
cd¿ ⫽
ltc
V ⫹ s¿m 11 ⫺ Vf 2
d f
(16.19)
where d is the fiber diameter and tc is the smaller of either the fiber–matrix bond
strength or the matrix shear yield strength.
Discontinuous and Randomly Oriented Fiber Composites
For a discontinuous
and randomly
oriented fiberreinforced
composite, modulus
of elasticity
Normally, when the fiber orientation is random, short and discontinuous fibers are
used; reinforcement of this type is schematically demonstrated in Figure 16.8c.
Under these circumstances, a “rule-of-mixtures” expression for the elastic modulus
similar to Equation 16.10a may be utilized, as follows:
Ecd ⫽ KEfVf ⫹ EmVm
(16.20)
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594 • Chapter 16 / Composites
Table 16.2 Properties of Unreinforced and Reinforced Polycarbonates with
Randomly Oriented Glass Fibers
Fiber Reinforcement (vol%)
Property
Unreinforced
20
30
40
Specific gravity
1.19–1.22
1.35
1.43
1.52
Tensile strength
[MPa (ksi)]
59–62
(8.5–9.0)
110
(16)
131
(19)
159
(23)
2.24–2.345
(0.325–0.340)
5.93
(0.86)
8.62
(1.25)
11.6
(1.68)
Elongation (%)
90–115
4–6
3–5
3–5
Impact strength,
notched Izod (lbf/in.)
12–16
2.0
2.0
2.5
Modulus of elasticity
[GPa (106 psi)]
Source: Adapted from Materials Engineering’s Materials Selector, copyright © Penton/IPC.
In this expression, K is a fiber efficiency parameter that depends on Vf and the
Ef ⲐEm ratio. Of course, its magnitude will be less than unity, usually in the range 0.1
to 0.6. Thus, for random fiber reinforcement (as with oriented), the modulus increases in some proportion of the volume fraction of fiber. Table 16.2, which gives
some of the mechanical properties of unreinforced and reinforced polycarbonates
for discontinuous and randomly oriented glass fibers, provides an idea of the magnitude of the reinforcement that is possible.
By way of summary, then, aligned fibrous composites are inherently anisotropic,
in that the maximum strength and reinforcement are achieved along the alignment
(longitudinal) direction. In the transverse direction, fiber reinforcement is virtually
nonexistent: fracture usually occurs at relatively low tensile stresses. For other stress
orientations, composite strength lies between these extremes. The efficiency of fiber
reinforcement for several situations is presented in Table 16.3; this efficiency is taken
to be unity for an oriented fiber composite in the alignment direction, and zero perpendicular to it.
When multidirectional stresses are imposed within a single plane, aligned layers that are fastened together one on top of another at different orientations are
frequently utilized. These are termed laminar composites, which are discussed in
Section 16.14.
Table 16.3 Reinforcement Efficiency of Fiber-Reinforced Composites for
Several Fiber Orientations and at Various Directions of Stress
Application
Fiber Orientation
Stress Direction
All fibers parallel
Parallel to fibers
Perpendicular to fibers
Any direction in the plane
of the fibers
Any direction
Fibers randomly and uniformly
distributed within a specific plane
Fibers randomly and uniformly
distributed within three
dimensions in space
Reinforcement
Efficiency
1
0
3
8
1
5
Source: H. Krenchel, Fibre Reinforcement, Copenhagen: Akademisk Forlag, 1964 [33].
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16.6 The Fiber Phase • 595
Applications involving totally multidirectional applied stresses normally use
discontinuous fibers, which are randomly oriented in the matrix material. Table
16.3 shows that the reinforcement efficiency is only one-fifth that of an aligned
composite in the longitudinal direction; however, the mechanical characteristics
are isotropic.
Consideration of orientation and fiber length for a particular composite will
depend on the level and nature of the applied stress as well as fabrication cost. Production rates for short-fiber composites (both aligned and randomly oriented) are
rapid, and intricate shapes can be formed that are not possible with continuous fiber
reinforcement. Furthermore, fabrication costs are considerably lower than for continuous and aligned; fabrication techniques applied to short-fiber composite materials include compression, injection, and extrusion molding, which are described for
unreinforced polymers in Section 15.22.
Concept Check 16.3
Cite one desirable characteristic and one less desirable characteristic for each of
(1) discontinuous-oriented, and (2) discontinuous-randomly oriented fiber-reinforced
composites.
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]
16.6 THE FIBER PHASE
whisker
fiber
An important characteristic of most materials, especially brittle ones, is that a smalldiameter fiber is much stronger than the bulk material. As discussed in Section 12.8,
the probability of the presence of a critical surface flaw that can lead to fracture
diminishes with decreasing specimen volume, and this feature is used to advantage
in the fiber-reinforced composites. Also, the materials used for reinforcing fibers
have high tensile strengths.
On the basis of diameter and character, fibers are grouped into three different classifications: whiskers, fibers, and wires. Whiskers are very thin single
crystals that have extremely large length-to-diameter ratios. As a consequence
of their small size, they have a high degree of crystalline perfection and are virtually flaw free, which accounts for their exceptionally high strengths; they are
among the strongest known materials. In spite of these high strengths, whiskers
are not utilized extensively as a reinforcement medium because they are extremely expensive. Moreover, it is difficult and often impractical to incorporate
whiskers into a matrix. Whisker materials include graphite, silicon carbide, silicon nitride, and aluminum oxide; some mechanical characteristics of these
materials are given in Table 16.4.
Materials that are classified as fibers are either polycrystalline or amorphous
and have small diameters; fibrous materials are generally either polymers or ceramics (e.g., the polymer aramids, glass, carbon, boron, aluminum oxide, and silicon
carbide). Table 16.4 also presents some data on a few materials that are used in
fiber form.
Fine wires have relatively large diameters; typical materials include steel,
molybdenum, and tungsten. Wires are utilized as a radial steel reinforcement in
automobile tires, in filament-wound rocket casings, and in wire-wound high-pressure
hoses.
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596 • Chapter 16 / Composites
Table 16.4 Characteristics of Several Fiber-Reinforcement Materials
Material
Specific Gravity
Graphite
2.2
Silicon nitride
3.2
Aluminum oxide
4.0
Silicon carbide
3.2
Aluminum oxide
3.95
Aramid (Kevlar 49™)
1.44
Carbona
1.78–2.15
E-glass
2.58
Boron
2.57
Silicon carbide
3.0
UHMWPE (Spectra 900™)
0.97
High-strength steel
7.9
Molybdenum
10.2
Tungsten
19.3
Tensile
Strength
[GPa (106 psi)]
Whiskers
20
(3)
5–7
(0.75–1.0)
10–20
(1–3)
20
(3)
Fibers
1.38
(0.2)
3.6–4.1
(0.525–0.600)
1.5–4.8
(0.22–0.70)
3.45
(0.5)
3.6
(0.52)
3.9
(0.57)
2.6
(0.38)
Metallic Wires
2.39
(0.35)
2.2
(0.32)
2.89
(0.42)
Specific
Strength
(GPa)
Modulus
of Elasticity
[GPa (106 psi)]
Specific
Modulus
(GPa)
9.1
700
(100)
350–380
(50–55)
700–1500
(100–220)
480
(70)
318
1.56–2.2
2.5–5.0
6.25
0.35
2.5–2.85
0.70–2.70
1.34
1.40
1.30
2.68
0.30
0.22
0.15
379
(55)
131
(19)
228–724
(32–100)
72.5
(10.5)
400
(60)
400
(60)
117
(17)
210
(30)
324
(47)
407
(59)
109–118
175–375
150
96
91
106–407
28.1
156
133
121
26.6
31.8
21.1
The term “carbon” instead of “graphite” is used to denote these fibers, since they are composed of crystalline
graphite regions, and also of noncrystalline material and areas of crystal misalignment.
a
16.7 THE MATRIX PHASE
matrix phase
The matrix phase of fibrous composites may be a metal, polymer, or ceramic. In
general, metals and polymers are used as matrix materials because some ductility
is desirable; for ceramic-matrix composites (Section 16.10), the reinforcing component is added to improve fracture toughness. The discussion of this section will focus
on polymer and metal matrices.
For fiber-reinforced composites, the matrix phase serves several functions. First,
it binds the fibers together and acts as the medium by which an externally applied
stress is transmitted and distributed to the fibers; only a very small proportion of
an applied load is sustained by the matrix phase. Furthermore, the matrix material
should be ductile. In addition, the elastic modulus of the fiber should be much higher
than that of the matrix. The second function of the matrix is to protect the individual fibers from surface damage as a result of mechanical abrasion or chemical
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16.8 Polymer-Matrix Composites • 597
reactions with the environment. Such interactions may introduce surface flaws capable of forming cracks, which may lead to failure at low tensile stress levels. Finally,
the matrix separates the fibers and, by virtue of its relative softness and plasticity,
prevents the propagation of brittle cracks from fiber to fiber, which could result in
catastrophic failure; in other words, the matrix phase serves as a barrier to crack
propagation. Even though some of the individual fibers fail, total composite fracture will not occur until large numbers of adjacent fibers, once having failed, form
a cluster of critical size.
It is essential that adhesive bonding forces between fiber and matrix be high
to minimize fiber pull-out. In fact, bonding strength is an important consideration
in the choice of the matrix–fiber combination. The ultimate strength of the composite depends to a large degree on the magnitude of this bond; adequate bonding
is essential to maximize the stress transmittance from the weak matrix to the strong
fibers.
16.8 POLYMER-MATRIX COMPOSITES
polymer-matrix
composite
Polymer-matrix composites (PMCs) consist of a polymer resin1 as the matrix, with
fibers as the reinforcement medium. These materials are used in the greatest
diversity of composite applications, as well as in the largest quantities, in light of
their room-temperature properties, ease of fabrication, and cost. In this section the
various classifications of PMCs are discussed according to reinforcement type (i.e.,
glass, carbon, and aramid), along with their applications and the various polymer
resins that are employed.
Glass Fiber-Reinforced Polymer (GFRP) Composites
Fiberglass is simply a composite consisting of glass fibers, either continuous or discontinuous, contained within a polymer matrix; this type of composite is produced
in the largest quantities. The composition of the glass that is most commonly drawn
into fibers (sometimes referred to as E-glass) is contained in Table 13.1; fiber
diameters normally range between 3 and 20 mm. Glass is popular as a fiber reinforcement material for several reasons:
1. It is easily drawn into high-strength fibers from the molten state.
2. It is readily available and may be fabricated into a glass-reinforced plastic
economically using a wide variety of composite-manufacturing techniques.
3. As a fiber it is relatively strong, and when embedded in a plastic matrix, it
produces a composite having a very high specific strength.
4. When coupled with the various plastics, it possesses a chemical inertness that
renders the composite useful in a variety of corrosive environments.
The surface characteristics of glass fibers are extremely important because
even minute surface flaws can deleteriously affect the tensile properties, as discussed in Section 12.8. Surface flaws are easily introduced by rubbing or abrading the surface with another hard material. Also, glass surfaces that have been
exposed to the normal atmosphere for even short time periods generally have a
weakened surface layer that interferes with bonding to the matrix. Newly drawn
fibers are normally coated during drawing with a “size,” a thin layer of a substance
1
The term “resin” is used in this context to denote a high-molecular-weight reinforcing
plastic.
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that protects the fiber surface from damage and undesirable environmental interactions. This size is ordinarily removed prior to composite fabrication and replaced
with a “coupling agent” or finish that produces a chemical bond between the fiber
and matrix.
There are several limitations to this group of materials. In spite of having high
strengths, they are not very stiff and do not display the rigidity that is necessary
for some applications (e.g., as structural members for airplanes and bridges). Most
fiberglass materials are limited to service temperatures below 200⬚C (400⬚F); at
higher temperatures, most polymers begin to flow or to deteriorate. Service temperatures may be extended to approximately 300⬚C (575⬚F) by using high-purity
fused silica for the fibers and high-temperature polymers such as the polyimide
resins.
Many fiberglass applications are familiar: automotive and marine bodies, plastic
pipes, storage containers, and industrial floorings. The transportation industries are
utilizing increasing amounts of glass fiber-reinforced plastics in an effort to decrease
vehicle weight and boost fuel efficiencies. A host of new applications are being used
or currently investigated by the automotive industry.
Carbon Fiber-Reinforced Polymer (CFRP) Composites
Carbon is a high-performance fiber material that is the most commonly used reinforcement in advanced (i.e., nonfiberglass) polymer-matrix composites. The reasons
for this are as follows:
1. Carbon fibers have the highest specific modulus and specific strength of all
reinforcing fiber materials.
2. They retain their high tensile modulus and high strength at elevated temperatures; high-temperature oxidation, however, may be a problem.
3. At room temperature, carbon fibers are not affected by moisture or a wide
variety of solvents, acids, and bases.
4. These fibers exhibit a diversity of physical and mechanical characteristics,
allowing composites incorporating these fibers to have specific engineered
properties.
5. Fiber and composite manufacturing processes have been developed that are
relatively inexpensive and cost effective.
Use of the term “carbon fiber” may seem perplexing since carbon is an element, and, as noted in Section 12.4, the stable form of crystalline carbon at ambient conditions is graphite, having the structure represented in Figure 12.17. Carbon
fibers are not totally crystalline, but are composed of both graphitic and noncrystalline regions; these areas of noncrystallinity are devoid of the three-dimensional
ordered arrangement of hexagonal carbon networks that is characteristic of graphite
(Figure 12.17).
Manufacturing techniques for producing carbon fibers are relatively complex
and will not be discussed. However, three different organic precursor materials are
used: rayon, polyacrylonitrile (PAN), and pitch. Processing technique will vary from
precursor to precursor, as will also the resultant fiber characteristics.
One classification scheme for carbon fibers is by tensile modulus; on this basis
the four classes are standard, intermediate, high, and ultrahigh moduli. Furthermore,
fiber diameters normally range between 4 and 10 mm; both continuous and chopped
forms are available. In addition, carbon fibers are normally coated with a protective
epoxy size that also improves adhesion with the polymer matrix.
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16.8 Polymer-Matrix Composites • 599
Carbon-reinforced polymer composites are currently being utilized extensively
in sports and recreational equipment (fishing rods, golf clubs), filament-wound
rocket motor cases, pressure vessels, and aircraft structural components—both military and commercial, fixed wing and helicopters (e.g., as wing, body, stabilizer, and
rudder components).
Aramid Fiber-Reinforced Polymer Composites
Aramid fibers are high-strength, high-modulus materials that were introduced in
the early 1970s. They are especially desirable for their outstanding strength-toweight ratios, which are superior to metals. Chemically, this group of materials is
known as poly(paraphenylene terephthalamide). There are a number of aramid
materials; trade names for two of the most common are Kevlar™ and Nomex™.
For the former, there are several grades (viz. Kevlar 29, 49, and 149) that have different mechanical behaviors. During synthesis, the rigid molecules are aligned in
the direction of the fiber axis, as liquid crystal domains (Section 15.19); the repeat
unit and the mode of chain alignment are represented in Figure 16.10. Mechanically, these fibers have longitudinal tensile strengths and tensile moduli (Table 16.4)
that are higher than other polymeric fiber materials; however, they are relatively
weak in compression. In addition, this material is known for its toughness, impact
resistance, and resistance to creep and fatigue failure. Even though the aramids are
thermoplastics, they are, nevertheless, resistant to combustion and stable to relatively high temperatures; the temperature range over which they retain their high
mechanical properties is between ⫺200 and 200⬚C (⫺330 and 390⬚F). Chemically,
they are susceptible to degradation by strong acids and bases, but they are relatively inert in other solvents and chemicals.
The aramid fibers are most often used in composites having polymer matrices;
common matrix materials are the epoxies and polyesters. Since the fibers are relatively
flexible and somewhat ductile, they may be processed by most common textile operations. Typical applications of these aramid composites are in ballistic products (bulletproof vests and armor), sporting goods, tires, ropes, missile cases, pressure vessels, and
as a replacement for asbestos in automotive brake and clutch linings, and gaskets.
The properties of continuous and aligned glass-, carbon-, and aramid-fiber reinforced epoxy composites are included in Table 16.5. Thus, a comparison of the mechanical characteristics of these three materials may be made in both longitudinal and
transverse directions.
Repeat unit
O
O
H
C
N
C
N
C
O
O
H
C
C
H
O
O
C
N
H
C
N
H
O
Fiber direction
N
C
O
Figure 16.10 Schematic
representation of repeat unit
and chain structures for aramid
(Kevlar) fibers. Chain alignment
with the fiber direction and
hydrogen bonds that form
between adjacent chains are also
shown. [From F. R. Jones
(Editor), Handbook of PolymerFibre Composites. Copyright ©
1994 by Addison-Wesley
Longman. Reprinted with
permission.]
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600 • Chapter 16 / Composites
Table 16.5 Properties of Continuous and Aligned Glass-, Carbon-, and
Aramid-Fiber Reinforced Epoxy-Matrix Composites in Longitudinal
and Transverse Directions. In All Cases the Fiber Volume Fraction
Is 0.60
Property
Specific gravity
Tensile modulus
Longitudinal [GPa (106 psi)]
Transverse [GPa (106 psi)]
Tensile strength
Longitudinal [MPa (ksi)]
Transverse [MPa (ksi)]
Ultimate tensile strain
Longitudinal
Transverse
Glass
(E-glass)
Carbon
(High Strength)
Aramid
(Kevlar 49)
2.1
1.6
1.4
45 (6.5)
12 (1.8)
145 (21)
10 (1.5)
76 (11)
5.5 (0.8)
1020 (150)
40 (5.8)
1240 (180)
41 (6)
1380 (200)
30 (4.3)
2.3
0.4
0.9
0.4
1.8
0.5
Source: Adapted from R. F. Floral and S. T. Peters, “Composite Structures and
Technologies,” tutorial notes, 1989.
Other Fiber Reinforcement Materials
Glass, carbon, and the aramids are the most common fiber reinforcements incorporated in polymer matrices. Other fiber materials that are used to much lesser degrees are boron, silicon carbide, and aluminum oxide; tensile moduli, tensile
strengths, specific strengths, and specific moduli of these materials in fiber form are
contained in Table 16.4. Boron fiber-reinforced polymer composites have been used
in military aircraft components, helicopter rotor blades, and some sporting goods.
Silicon carbide and aluminum oxide fibers are utilized in tennis rackets, circuit
boards, military armor, and rocket nose cones.
Polymer Matrix Materials
The roles assumed by the polymer matrix are outlined in Section 16.7. In addition, the matrix often determines the maximum service temperature, since it
normally softens, melts, or degrades at a much lower temperature than the fiber
reinforcement.
The most widely utilized and least expensive polymer resins are the polyesters
and vinyl esters;2 these matrix materials are used primarily for glass fiber-reinforced
composites.A large number of resin formulations provide a wide range of properties
for these polymers. The epoxies are more expensive and, in addition to commercial
applications, are also utilized extensively in PMCs for aerospace applications; they
have better mechanical properties and resistance to moisture than the polyesters
and vinyl resins. For high-temperature applications, polyimide resins are employed;
their continuous-use, upper-temperature limit is approximately 230⬚C (450⬚F). Finally, high-temperature thermoplastic resins offer the potential to be used in future
aerospace applications; such materials include polyetheretherketone (PEEK),
poly(phenylene sulfide) (PPS), and polyetherimide (PEI).
2
The chemistry and typical properties of some of the matrix materials discussed in this
section are included in Appendices B, D, and E.
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16.8 Polymer-Matrix Composites • 601
DESIGN EXAMPLE 16.1
Design of a Tubular Composite Shaft
A tubular composite shaft is to be designed that has an outside diameter of
70 mm (2.75 in.), an inside diameter of 50 mm (1.97 in.) and a length of 1.0 m
(39.4 in.); such is represented schematically in Figure 16.11. The mechanical
characteristic of prime importance is bending stiffness in terms of the longitudinal modulus of elasticity; strength and fatigue resistance are not significant
parameters for this application when filament composites are utilized. Stiffness
is to be specified as maximum allowable deflection in bending; when subjected
to three-point bending as in Figure 12.32 (i.e., support points at both tube extremities and load application at the longitudinal midpoint), a load of 1000 N
(225 lbf) is to produce an elastic deflection of no more than 0.35 mm (0.014 in.)
at the midpoint position.
Continuous fibers that are oriented parallel to the tube axis will be used;
possible fiber materials are glass, and carbon in standard-, intermediate-, and highmodulus grades. The matrix material is to be an epoxy resin, and the maximum
allowable fiber volume fraction is 0.60.
This design problem calls for us to do the following:
(a) Decide which of the four fiber materials, when embedded in the epoxy matrix,
meet the stipulated criteria.
(b) Of these possibilities, select the one fiber material that will yield the
lowest-cost composite material (assuming fabrication costs are the same for
all fibers).
Elastic modulus, density, and cost data for the fiber and matrix materials are
contained in Table 16.6.
Solution
(a) It first becomes necessary to determine the required longitudinal modulus of elasticity for this composite material, consistent with the stipulated
criteria. This computation necessitates the use of the three-point deflection
expression
FL3
(16.21)
¢y ⫽
48 EI
in which ¢y is the midpoint deflection, F is the applied force, L is the support
point separation distance, E is the modulus of elasticity, and I is the cross-sectional
50 70
mm mm
1.0 m
Figure 16.11 Schematic representation of a tubular composite shaft, the subject of
Design Example 16.1.
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602 • Chapter 16 / Composites
Table 16.6 Elastic Modulus, Density, and Cost Data for Glass and
Various Carbon Fibers, and Epoxy Resin
Material
Elastic Modulus
(GPa)
Density
(g/cm3)
Cost
($/kg)
72.5
230
2.58
1.80
2.50
35.00
285
1.80
70.00
400
1.80
175.00
1.14
9.00
Glass fibers
Carbon fibers
(standard modulus)
Carbon fibers
(intermediate modulus)
Carbon fibers
(high modulus)
Epoxy resin
2.4
moment of inertia. For a tube having inside and outside diameters of di and do,
respectively,
I⫽
p
1d 4 ⫺ d 4i 2
64 o
(16.22)
and
E⫽
For this shaft design,
4FL3
3p¢y 1d 4o ⫺ d 4i 2
(16.23)
F ⫽ 1000 N
L ⫽ 1.0 m
¢y ⫽ 0.35 mm
do ⫽ 70 mm
di ⫽ 50 mm
Thus, the required longitudinal modulus of elasticity for this shaft is
E⫽
411000 N2 11.0 m2 3
3p10.35 ⫻ 10⫺3 m2 3 170 ⫻ 10⫺3 m2 4 ⫺ 150 ⫻ 10⫺3 m2 4 4
⫽ 69.3 GPa 19.9 ⫻ 106 psi2
The next step is to determine the fiber and matrix volume fractions for each
of the four candidate fiber materials. This is possible using the rule-of-mixtures
expression, Equation 16.10b:
Ecs ⫽ EmVm ⫹ EfVf ⫽ Em 11 ⫺ Vf 2 ⫹ EfVf
In Table 16.7 is given a tabulation of the Vm and Vf values required for Ecs ⫽ 69.3
GPa; Equation 16.10b and the moduli data in Table 16.6 were used in these computations. Here it may be noted that only the three carbon fiber types are possible candidates since their Vf values are less than 0.6.
(b) At this point it becomes necessary to determine the volume of fibers
and matrix for each of the three carbon types. The total tube volume Vc in
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16.9 Metal-Matrix Composites • 603
centimeters is
pL 2
1do ⫺ di2 2
4
p1100 cm2
⫽
3 17.0 cm2 2 ⫺ 15.0 cm2 2 4
4
⫽ 1885 cm3 1114 in.3 2
(16.24)
Vc ⫽
Table 16.7 Fiber and Matrix Volume Fractions
for Three Carbon Fiber Types as
Required to Give a Composite
Modulus of 69.3 GPa
Fiber Type
Glass
Carbon
(standard modulus)
Carbon
(intermediate modulus)
Carbon
(high modulus)
Vf
Vm
0.954
0.293
0.046
0.707
0.237
0.763
0.168
0.832
Thus, fiber and matrix volumes result from products of this value and the Vf and
Vm values cited in Table 16.7. These volume values are presented in Table 16.8,
which are then converted into masses using densities (Table 16.6), and finally
into material costs, from the per unit mass cost (also given in Table 16.6).
As may be noted in Table 16.8, the material of choice (i.e., the least expensive) is the standard-modulus carbon-fiber composite; the relatively low cost per
unit mass of this fiber material offsets its relatively low modulus of elasticity and
required high volume fraction.
Table 16.8 Fiber and Matrix Volumes, Masses, and Costs and Total Material
Cost for Three Carbon Fiber-Epoxy-Matrix Composites
Fiber Type
Carbon
(standard
modulus)
Carbon
(intermediate
modulus)
Carbon
(high modulus)
Fiber
Volume
(cm3)
Fiber
Mass
(kg)
Fiber
Cost
($)
Matrix
Volume
(cm3)
Matrix
Mass
(kg)
Matrix
Cost
($)
Total
Cost
($)
552
0.994
34.80
1333
1.520
13.70
48.50
447
0.805
56.35
1438
1.639
14.75
71.10
317
0.571
99.90
1568
1.788
16.10
116.00
16.9 METAL-MATRIX COMPOSITES
metal-matrix
composite
As the name implies, for metal-matrix composites (MMCs) the matrix is a ductile
metal. These materials may be utilized at higher service temperatures than their
base metal counterparts; furthermore, the reinforcement may improve specific
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604 • Chapter 16 / Composites
stiffness, specific strength, abrasion resistance, creep resistance, thermal conductivity, and dimensional stability. Some of the advantages of these materials over the
polymer-matrix composites include higher operating temperatures, nonflammability,
and greater resistance to degradation by organic fluids. Metal-matrix composites
are much more expensive than PMCs, and, therefore, their (MMC) use is somewhat
restricted.
The superalloys, as well as alloys of aluminum, magnesium, titanium, and copper, are employed as matrix materials. The reinforcement may be in the form of
particulates, both continuous and discontinuous fibers, and whiskers; concentrations
normally range between 10 and 60 vol%. Continuous fiber materials include carbon, silicon carbide, boron, aluminum oxide, and the refractory metals. On the other
hand, discontinuous reinforcements consist primarily of silicon carbide whiskers,
chopped fibers of aluminum oxide and carbon, and particulates of silicon carbide
and aluminum oxide. In a sense, the cermets (Section 16.2) fall within this MMC
scheme. In Table 16.9 are presented the properties of several common metal-matrix,
continuous and aligned fiber-reinforced composites.
Some matrix–reinforcement combinations are highly reactive at elevated temperatures. Consequently, composite degradation may be caused by high-temperature
processing or by subjecting the MMC to elevated temperatures during service. This
problem is commonly resolved either by applying a protective surface coating to
the reinforcement or by modifying the matrix alloy composition.
Normally the processing of MMCs involves at least two steps: consolidation or
synthesis (i.e., introduction of reinforcement into the matrix), followed by a shaping
operation. A host of consolidation techniques are available, some of which are relatively sophisticated; discontinuous fiber MMCs are amenable to shaping by standard
metal-forming operations (e.g., forging, extrusion, rolling).
Automobile manufacturers have recently begun to use MMCs in their products. For example, some engine components have been introduced consisting of an
aluminum-alloy matrix that is reinforced with aluminum oxide and carbon fibers;
this MMC is light in weight and resists wear and thermal distortion. Metal-matrix
composites are also employed in driveshafts (that have higher rotational speeds and
reduced vibrational noise levels), extruded stabilizer bars, and forged suspension
and transmission components.
The aerospace industry also uses MMCs. Structural applications include advanced
aluminum alloy metal-matrix composites; boron fibers are used as the reinforcement for the Space Shuttle Orbiter, and continuous graphite fibers for the Hubble
Telescope.
Table 16.9 Properties of Several Metal-Matrix Composites Reinforced with Continuous and
Aligned Fibers
Fiber
Carbon
Boron
SiC
Alumina
Carbon
Borsic
Matrix
Fiber Content
(vol%)
Density
(g/cm3)
Longitudinal Tensile
Modulus (GPa)
Longitudinal Tensile
Strength (MPa)
6061 Al
6061 Al
6061 Al
380.0 Al
AZ31 Mg
Ti
41
48
50
24
38
45
2.44
—
2.93
—
1.83
3.68
320
207
230
120
300
220
620
1515
1480
340
510
1270
Source: Adapted from J. W. Weeton, D. M. Peters, and K. L. Thomas, Engineers’ Guide to Composite Materials,
ASM International, Materials Park, OH, 1987.
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16.10 Ceramic-Matrix Composites • 605
The high-temperature creep and rupture properties of some of the superalloys
(Ni- and Co-based alloys) may be enhanced by fiber reinforcement using refractory metals such as tungsten. Excellent high-temperature oxidation resistance and
impact strength are also maintained. Designs incorporating these composites permit higher operating temperatures and better efficiencies for turbine engines.
16.10 CERAMIC-MATRIX COMPOSITES
ceramic-matrix
composite
As discussed in Chapters 12 and 13, ceramic materials are inherently resilient to
oxidation and deterioration at elevated temperatures; were it not for their disposition to brittle fracture, some of these materials would be ideal candidates for use
in high-temperature and severe-stress applications, specifically for components in
automobile and aircraft gas turbine engines. Fracture toughness values for ceramic
materials are low and typically lie between 1 and 5 MPa 1m (0.9 and 4.5 ksi 1in.),
Table 8.1 and Table B.5, Appendix B. By way of contrast, KIc values for most metals are much higher (15 to greater than 150 MPa 1m [14 to 7 140 ksi 1in.])
The fracture toughnesses of ceramics have been improved significantly by
the development of a new generation of ceramic-matrix composites (CMCs)—
particulates, fibers, or whiskers of one ceramic material that have been embedded
into a matrix of another ceramic. Ceramic-matrix composite materials have extended
fracture toughnesses to between about 6 and 20 MPa 1m (5.5 and 18 ksi 1in.).
In essence, this improvement in the fracture properties results from interactions
between advancing cracks and dispersed phase particles. Crack initiation normally
occurs with the matrix phase, whereas crack propagation is impeded or hindered
by the particles, fibers, or whiskers. Several techniques are utilized to retard crack
propagation, which are discussed as follows.
One particularly interesting and promising toughening technique employs a
phase transformation to arrest the propagation of cracks and is aptly termed transformation toughening. Small particles of partially stabilized zirconia (Section 12.7)
are dispersed within the matrix material, often Al2O3 or ZrO2 itself. Typically, CaO,
MgO, Y2O3, and CeO are used as stabilizers. Partial stabilization allows retention
of the metastable tetragonal phase at ambient conditions rather than the stable
monoclinic phase; these two phases are noted on the ZrO2–ZrCaO3 phase diagram,
Figure 12.26. The stress field in front of a propagating crack causes these metastably
retained tetragonal particles to undergo transformation to the stable monoclinic
phase. Accompanying this transformation is a slight particle volume increase, and
the net result is that compressive stresses are established on the crack surfaces near
the crack tip that tend to pinch the crack shut, thereby arresting its growth. This
process is demonstrated schematically in Figure 16.12.
Other recently developed toughening techniques involve the utilization of ceramic whiskers, often SiC or Si3N4. These whiskers may inhibit crack propagation
by (1) deflecting crack tips, (2) forming bridges across crack faces, (3) absorbing energy during pull-out as the whiskers debond from the matrix, and/or (4) causing a
redistribution of stresses in regions adjacent to the crack tips.
In general, increasing fiber content improves strength and fracture toughness;
this is demonstrated in Table 16.10 for SiC whisker-reinforced alumina. Furthermore, there is a considerable reduction in the scatter of fracture strengths for
whisker-reinforced ceramics relative to their unreinforced counterparts. In addition,
these CMCs exhibit improved high-temperature creep behavior and resistance to
thermal shock (i.e., failure resulting from sudden changes in temperature).
Ceramic-matrix composites may be fabricated using hot pressing, hot isostatic
pressing, and liquid phase sintering techniques. Relative to applications, SiC
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Monoclinic
ZrO2
particles
Figure 16.12
Schematic
demonstration of
transformation
toughening. (a) A
crack prior to
inducement of the
ZrO2 particle phase
transformation.
(b) Crack arrestment
due to the stressinduced phase
transformation.
Stress
field
region
Crack
Crack
(a)
Tetragonal
ZrO2
particles
Tetragonal
ZrO2
particles
(b)
whisker-reinforced aluminas are being utilized as cutting tool inserts for machining hard metal alloys; tool lives for these materials are greater than for cemented
carbides (Section 16.2).
16.11 CARBON–CARBON COMPOSITES
carbon–carbon
composite
One of the most advanced and promising engineering material is the carbon fiberreinforced carbon-matrix composite, often termed a carbon–carbon composite; as
the name implies, both reinforcement and matrix are carbon. These materials are
relatively new and expensive and, therefore, are not currently being utilized extensively. Their desirable properties include high-tensile moduli and tensile strengths
that are retained to temperatures in excess of 2000⬚C (3630⬚F), resistance to creep,
and relatively large fracture toughness values. Furthermore, carbon–carbon composites have low coefficients of thermal expansion and relatively high thermal conductivities; these characteristics, coupled with high strengths, give rise to a relatively
low susceptibility to thermal shock. Their major drawback is a propensity to hightemperature oxidation.
The carbon–carbon composites are employed in rocket motors, as friction materials in aircraft and high-performance automobiles, for hot-pressing molds, in components for advanced turbine engines, and as ablative shields for re-entry vehicles.
Table 16.10 Room Temperature Fracture Strengths and
Fracture Toughnesses for Various SiC Whisker
Contents in Al2O3
Whisker
Content (vol%)
Fracture
Strength (MPa)
Fracture Toughness
(MPa 1m)
0
10
20
40
—
455 ⫾ 55
655 ⫾ 135
850 ⫾ 130
4.5
7.1
7.5–9.0
6.0
Source: Adapted from Engineered Materials Handbook, Vol. 1,
Composites, C. A. Dostal (Senior Editor), ASM International,
Materials Park, OH, 1987.
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16.13 Processing of Fiber-Reinforced Composites • 607
The primary reason that these composite materials are so expensive is the
relatively complex processing techniques that are employed. Preliminary procedures are similar to those used for carbon-fiber, polymer-matrix composites. That
is, the continuous carbon fibers are laid down having the desired two- or threedimensional pattern; these fibers are then impregnated with a liquid polymer
resin, often a phenolic; the workpiece is next formed into the final shape, and
the resin is allowed to cure. At this time the matrix resin is “pyrolyzed,” that is,
converted into carbon by heating in an inert atmosphere; during pyrolysis, molecular components consisting of oxygen, hydrogen, and nitrogen are driven off,
leaving behind large carbon chain molecules. Subsequent heat treatments at
higher temperatures will cause this carbon matrix to densify and increase in
strength. The resulting composite, then, consists of the original carbon fibers that
remained essentially unaltered, which are contained in this pyrolyzed carbon
matrix.
16.12 HYBRID COMPOSITES
hybrid composite
A relatively new fiber-reinforced composite is the hybrid, which is obtained by using
two or more different kinds of fibers in a single matrix; hybrids have a better allaround combination of properties than composites containing only a single fiber
type. A variety of fiber combinations and matrix materials are used, but in the most
common system, both carbon and glass fibers are incorporated into a polymeric
resin. The carbon fibers are strong and relatively stiff and provide a low-density reinforcement; however, they are expensive. Glass fibers are inexpensive and lack the
stiffness of carbon. The glass–carbon hybrid is stronger and tougher, has a higher
impact resistance, and may be produced at a lower cost than either of the comparable all-carbon or all-glass reinforced plastics.
There are a number of ways in which the two different fibers may be combined,
which will ultimately affect the overall properties. For example, the fibers may all
be aligned and intimately mixed with one another; or laminations may be constructed consisting of layers, each of which consists of a single fiber type, alternating
one with another. In virtually all hybrids the properties are anisotropic.
When hybrid composites are stressed in tension, failure is usually noncatastrophic (i.e., does not occur suddenly). The carbon fibers are the first to fail, at
which time the load is transferred to the glass fibers. Upon failure of the glass fibers,
the matrix phase must sustain the applied load. Eventual composite failure concurs
with that of the matrix phase.
Principal applications for hybrid composites are lightweight land, water, and
air transport structural components, sporting goods, and lightweight orthopedic
components.
16.13 PROCESSING OF FIBER-REINFORCED
COMPOSITES
To fabricate continuous fiber-reinforced plastics that meet design specifications, the
fibers should be uniformly distributed within the plastic matrix and, in most instances, all oriented in virtually the same direction. In this section several techniques
(pultrusion, filament winding, and prepreg production processes) by which useful
products of these materials are manufactured will be discussed.
Pultrusion
Pultrusion is used for the manufacture of components having continuous lengths
and a constant cross-sectional shape (i.e., rods, tubes, beams, etc.). With this technique,
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Preforming
die
Curing
die
Pullers
Fiber
rovings
Resin
impregnation
tank
Figure 16.13 Schematic diagram showing the pultrusion process.
illustrated schematically in Figure 16.13, continuous fiber rovings, or tows,3 are first
impregnated with a thermosetting resin; these are then pulled through a steel die
that preforms to the desired shape and also establishes the resin/fiber ratio. The
stock then passes through a curing die that is precision machined so as to impart
the final shape; this die is also heated to initiate curing of the resin matrix. A pulling
device draws the stock through the dies and also determines the production speed.
Tubes and hollow sections are made possible by using center mandrels or inserted
hollow cores. Principal reinforcements are glass, carbon, and aramid fibers, normally
added in concentrations between 40 and 70 vol%. Commonly used matrix materials include polyesters, vinyl esters, and epoxy resins.
Pultrusion is a continuous process that is easily automated; production rates are
relatively high, making it very cost effective. Furthermore, a wide variety of shapes
are possible, and there is really no practical limit to the length of stock that may be
manufactured.
Prepreg Production Processes
prepreg
Prepreg is the composite industry’s term for continuous fiber reinforcement preimpregnated with a polymer resin that is only partially cured. This material is delivered
in tape form to the manufacturer, who then directly molds and fully cures the product without having to add any resin. It is probably the composite material form most
widely used for structural applications.
The prepregging process, represented schematically for thermoset polymers in
Figure 16.14, begins by collimating a series of spool-wound continuous fiber tows.
These tows are then sandwiched and pressed between sheets of release and carrier
paper using heated rollers, a process termed “calendering.” The release paper sheet
has been coated with a thin film of heated resin solution of relatively low viscosity
so as to provide for its thorough impregnation of the fibers. A “doctor blade” spreads
the resin into a film of uniform thickness and width. The final prepreg product—the
thin tape consisting of continuous and aligned fibers embedded in a partially cured
resin—is prepared for packaging by winding onto a cardboard core. As shown in Figure 16.14, the release paper sheet is removed as the impregnated tape is spooled.
Typical tape thicknesses range between 0.08 and 0.25 mm (3 ⫻ 10⫺3 and 10⫺2 in.),
tape widths range between 25 and 1525 mm (1 and 60 in.), whereas resin content
usually lies between about 35 and 45 vol%.
3
A roving, or tow, is a loose and untwisted bundle of continuous fibers that are drawn
together as parallel strands.
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16.13 Processing of Fiber-Reinforced Composites • 609
Hopper containing
heated resin
Figure 16.14
Schematic diagram
illustrating the
production of
prepreg tape using a
thermoset polymer.
Doctor
blade
Release
paper
Waste release
paper
Spooled
fiber
Heated calender
rolls
Carrier
paper
Spooled
prepreg
At room temperature the thermoset matrix undergoes curing reactions; therefore, the prepreg is stored at 0⬚C (32⬚F) or lower. Also, the time in use at room temperature (or “out-time”) must be minimized. If properly handled, thermoset prepregs
have a lifetime of at least six months and usually longer.
Both thermoplastic and thermosetting resins are utilized; carbon, glass, and
aramid fibers are the common reinforcements.
Actual fabrication begins with the “lay-up”—laying of the prepreg tape onto a
tooled surface. Normally a number of plies are laid up (after removal from the carrier backing paper) to provide the desired thickness. The lay-up arrangement may
be unidirectional, but more often the fiber orientation is alternated to produce a
cross-ply or angle-ply laminate. Final curing is accomplished by the simultaneous
application of heat and pressure.
The lay-up procedure may be carried out entirely by hand (hand lay-up), wherein
the operator both cuts the lengths of tape and then positions them in the desired
orientation on the tooled surface. Alternately, tape patterns may be machine cut,
then hand laid. Fabrication costs can be further reduced by automation of prepreg
lay-up and other manufacturing procedures (e.g., filament winding, as discussed below), which virtually eliminates the need for hand labor. These automated methods
are essential for many applications of composite materials to be cost effective.
Filament Winding
Filament winding is a process by which continuous reinforcing fibers are accurately
positioned in a predetermined pattern to form a hollow (usually cylindrical) shape.
The fibers, either as individual strands or as tows, are first fed through a resin bath
and then are continuously wound onto a mandrel, usually using automated winding equipment (Figure 16.15). After the appropriate number of layers have been
applied, curing is carried out either in an oven or at room temperature, after which
the mandrel is removed. As an alternative, narrow and thin prepregs (i.e., tow pregs)
10 mm or less in width may be filament wound.
Various winding patterns are possible (i.e., circumferential, helical, and polar)
to give the desired mechanical characteristics. Filament-wound parts have very high
strength-to-weight ratios. Also, a high degree of control over winding uniformity
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610 • Chapter 16 / Composites
Figure 16.15 Schematic representations of
helical, circumferential, and polar filament
winding techniques. [From N. L. Hancox,
(Editor), Fibre Composite Hybrid Materials,
The Macmillan Company, New York, 1981.]
Helical winding
Circumferential winding
Polar winding
and orientation is afforded with this technique. Furthermore, when automated, the
process is most economically attractive. Common filament-wound structures include
rocket motor casings, storage tanks and pipes, and pressure vessels.
Manufacturing techniques are now being used to produce a wide variety of
structural shapes that are not necessarily limited to surfaces of revolution (e.g.,
I-beams). This technology is advancing very rapidly because it is very cost effective.
St r u c t u r a l C o m p o s i t e s
structural composite
A structural composite is normally composed of both homogeneous and composite materials, the properties of which depend not only on the properties of the
constituent materials but also on the geometrical design of the various structural
elements. Laminar composites and sandwich panels are two of the most common
structural composites; only a relatively superficial examination is offered here for
them.
16.14 LAMINAR COMPOSITES
laminar composite
A laminar composite is composed of two-dimensional sheets or panels that have a
preferred high-strength direction such as is found in wood and continuous and
aligned fiber-reinforced plastics. The layers are stacked and subsequently cemented
together such that the orientation of the high-strength direction varies with each
successive layer (Figure 16.16). For example, adjacent wood sheets in plywood are
aligned with the grain direction at right angles to each other. Laminations may also
be constructed using fabric material such as cotton, paper, or woven glass fibers embedded in a plastic matrix. Thus a laminar composite has relatively high strength in
a number of directions in the two-dimensional plane; however, the strength in any
given direction is, of course, lower than it would be if all the fibers were oriented
in that direction. One example of a relatively complex laminated structure is the
modern ski (see the chapter-opening illustration for this chapter).
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16.15 Sandwich Panels • 611
Figure 16.16 The stacking of successive
oriented, fiber-reinforced layers for a laminar
composite.
16.15 SANDWICH PANELS
sandwich panel
Sandwich panels, considered to be a class of structural composites, are designed to
be light-weight beams or panels having relatively high stiffnesses and strengths. A
sandwich panel consists of two outer sheets, or faces, that are separated by and adhesively bonded to a thicker core (Figure 16.17). The outer sheets are made of a
relatively stiff and strong material, typically aluminum alloys, fiber-reinforced plastics, titanium, steel, or plywood; they impart high stiffness and strength to the structure, and must be thick enough to withstand tensile and compressive stresses that
result from loading.The core material is lightweight, and normally has a low modulus
of elasticity. Core materials typically fall within three categories: rigid polymeric
foams (i.e., phenolics, epoxy, polyurethanes), wood (i.e., balsa wood), and honeycombs
(see below).
Structurally, the core serves several functions. First of all, it provides continuous
support for the faces. In addition, it must have sufficient shear strength to withstand
transverse shear stresses, and also be thick enough to provide high shear stiffness
(to resist buckling of the panel). (It should be noted that tensile and compressive
stresses on the core are much lower than on the faces.)
Another popular core consists of a “honeycomb” structure—thin foils that have
been formed into interlocking hexagonal cells, with axes oriented perpendicular to
Figure 16.17
Schematic diagram
showing the cross
section of a sandwich
panel.
Transverse
direction
Faces
Core
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612 • Chapter 16 / Composites
Face sheet
Honeycomb
Adhesive
Face sheet
Fabricated
sandwich
panel
Figure 16.18 Schematic diagram showing the construction of a honeycomb core sandwich
panel. (Reprinted with permission from Engineered Materials Handbook, Vol. 1, Composites,
ASM International, Metals Park, OH, 1987.)
the face planes; Figure 16.18 shows a cutaway view of a honeycomb core sandwich
panel. The honeycomb material is normally either an aluminum alloy or aramid
polymer. Strength and stiffness of honeycomb structures depend on cell size, cell
wall thickness, and the material from which the honeycomb is made.
Sandwich panels are used in a wide variety of applications including roofs, floors,
and walls of buildings; and, in aerospace and aircraft (i.e., for wings, fuselage, and
tailplane skins).
MATERIALS OF IMPORTANCE
Nanocomposites in Tennis Balls
N
anocomposites—composites that consist of
nanosized particles embedded in some type
of matrix—are a group of promising new materials, that will undoubtedly become infused with
some of our modern technologies. In fact, one type
of nanocomposite is currently being used in highperformance tennis balls. These balls retain their
original pressure and bounce twice as long as conventional ones. Air permeation through the walls
of the ball is inhibited by a factor of two due to the
presence of a flexible and very thin (10 to 50 mm)
nanocomposite barrier coating that covers the inner core;4 a schematic diagram of the cross-section
of one of these tennis balls is shown in Figure 16.19.
Because of their outstanding characteristics, these
Double Core™ balls have recently been selected
as the official balls for some of the major tennis
tournaments.
This nanocomposite coating consists of a matrix of butyl rubber, within which is embedded thin
platelets of vermiculite,5 a natural clay mineral.
The vermiculite platelets exist as single-molecule
thin sheets—on the order of a nanometer thick—
that have a very large aspect ratio (of about
10,000); aspect ratio is the ratio of the lateral dimensions of a platelet to its thickness. Furthermore,
4
This coating was developed by InMat Inc., and is called Air D-Fense™. Wilson Sporting Goods has incorporated this coating in its Double Core™ tennis balls.
5
Vermiculite is one member of the layered silicates group that is discussed in Section 12.3.
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Summary • 613
the vermiculite platelets are exfoliated—that is,
they remain separated from one another. Also,
within the butyl rubber, the vermiculite platelets
are aligned such that all their lateral axes lie in the
same plane; and throughout this barrier coating
there are multiple layers of these platelets (per the
inset of Figure 16.19).
The presence of the vermiculite platelets accounts for the ability of the nanocomposite coating
to more effectively retain air pressure within the
tennis balls. These platelets act as multi-layer barriers to the diffusion of air molecules, and slow
down the diffusion rate; that is, the diffusion path
length of air molecules is enhanced significantly
since they (the air molecules) must bypass these
particles as they diffuse through the coating. Also,
the addition of the particles to the butyl rubber
does not diminish its flexibility.
It is anticipated that this type of coating can
also be applied to other kinds of sporting equipment (i.e., soccer balls, footballs, bicycle tires), as
well as to automobile tires (which would be lighter
in weight and more recyclable).
Vermiculite platelet
Butyl rubber
Cover
Outer core
Pressurized
air
Nanocomposite
barrier core
Figure 16.19 Schematic diagram showing the crosssection of a high-performance Double Core™ tennis
ball. The inset drawing presents a detailed view of the
nanocomposite coating that acts as a barrier to air
permeation.
Photograph of a can of Double Core™ tennis balls
and an individual ball. (Photograph courtesy of
Wilson Sporting Goods Company.)
SUMMARY
Introduction
Composites are artificially produced multiphase materials having a desirable combination of the best properties of the constituent phases. Usually, one phase (the
matrix) is continuous and completely surrounds the other (the dispersed phase). In
this discussion, composites were classified as particle-reinforced, fiber-reinforced,
and structural.
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614 • Chapter 16 / Composites
Large-Particle Composites
Dispersion-Strengthened Composites
Large-particle and dispersion-strengthened composites fall within the particlereinforced classification. For dispersion strengthening, improved strength is
achieved by extremely small particles of the dispersed phase, which inhibit dislocation motion; that is, the strengthening mechanism involves interactions that
may be treated on the atomic level. The particle size is normally greater with
large-particle composites, whose mechanical characteristics are enhanced by
reinforcement action.
Concrete, a type of large-particle composite, consists of an aggregate of particles
bonded together with cement. In the case of portland cement concrete, the aggregate
consists of sand and gravel; the cementitious bond develops as a result of chemical
reactions between the portland cement and water. The mechanical strength of this
concrete may be improved by reinforcement methods (e.g., embedment into the fresh
concrete of steel rods, wires, etc.). Additional reinforcement is possible by the imposition of residual compressive stresses using prestressing and posttensioning techniques.
Influence of Fiber Length
The Fiber Phase
Of the several composite types, the potential for reinforcement efficiency is greatest for those that are fiber reinforced. With these composites an applied load is
transmitted to and distributed among the fibers via the matrix phase, which in most
cases is at least moderately ductile. Significant reinforcement is possible only if the
matrix–fiber bond is strong. On the basis of diameter, fiber reinforcements are classified as whiskers, fibers, or wires. Since reinforcement discontinues at the fiber extremities, reinforcement efficiency depends on fiber length. For each fiber–matrix
combination, there exists some critical length; the length of continuous fibers greatly
exceeds this critical value, whereas shorter fibers are discontinuous.
Influence of Fiber Orientation and Concentration
Fiber arrangement is also crucial relative to composite characteristics. The mechanical properties of continuous and aligned fiber composites are highly anisotropic. In
the alignment direction, reinforcement and strength are a maximum; perpendicular
to the alignment, they are a minimum. The stress–strain behavior for longitudinal
loading was discussed. Composite rule-of-mixture expressions for the modulus in
both longitudinal and transverse orientations were developed; in addition, an equation
for longitudinal strength was also cited.
For short and discontinuous fibrous composites, the fibers may be either aligned
or randomly oriented. Significant strengths and stiffnesses are possible for aligned
short-fiber composites in the longitudinal direction. Despite some limitations on reinforcement efficiency, the properties of randomly oriented short-fiber composites
are isotropic.
Polymer-Matrix Composites
Metal-Matrix Composites
Ceramic-Matrix Composites
Carbon-Carbon Composites
Hybrid Composites
Fibrous-reinforced composites are sometimes classified according to matrix type;
within this scheme are three classifications: polymer-, metal-, and ceramic-matrix.
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Important Terms and Concepts • 615
Polymer-matrix are the most common, which may be reinforced with glass, carbon, and aramid fibers. Service temperatures are higher for metal-matrix composites, which also utilize a variety of fiber and whisker types. The objective of
many polymer- and metal-matrix composites is a high specific strength and/or
specific modulus, which requires matrix materials having low densities. With
ceramic-matrix composites, the design goal is increased fracture toughness. This
is achieved by interactions between advancing cracks and dispersed phase particles; transformation toughening is one such technique for improving Klc. Other
more advanced composites are carbon–carbon (carbon fibers embedded in a
pyrolyzed carbon matrix) and the hybrids (containing at least two different fiber
types).
Processing of Fiber-Reinforced Composites
Several composite processing techniques have been developed that provide a uniform fiber distribution and a high degree of alignment. With pultrusion, components
of continuous length and constant cross section are formed as resin-impregnated
fiber tows are pulled through a die. Composites utilized for many structural applications are commonly prepared using a lay-up operation (either hand or automated),
wherein prepreg tape plies are laid down on a tooled surface and are subsequently
fully cured by the simultaneous application of heat and pressure. Some hollow structures may be fabricated using automated filament winding procedures, whereby
resin-coated strands or tows or prepreg tape are continuously wound onto a mandrel,
followed by a curing operation.
Laminar Composites
Sandwich Panels
Two general kinds of structural composites were discussed: the laminar composites and sandwich panels. The properties of laminar composites are virtually
isotropic in a two-dimensional plane. This is made possible with several sheets of
a highly anisotropic composite, which are cemented onto one another such that
the high-strength direction is varied with each successive layer. Sandwich panels
consist of two strong and stiff sheet faces that are separated by a core material or
structure. These structures combine relatively high strengths and stiffnesses with
low densities.
I M P O R TA N T T E R M S A N D C O N C E P T S
Carbon–carbon composite
Ceramic-matrix composite
Cermet
Concrete
Dispersed phase
Dispersion-strengthened composite
Fiber
Fiber-reinforced composite
Hybrid composite
Laminar composite
Large-particle composite
Longitudinal direction
Matrix phase
Metal-matrix composite
Polymer-matrix composite
Prepreg
Prestressed concrete
Principle of combined action
Reinforced concrete
Rule of mixtures
Sandwich panel
Specific modulus
Specific strength
Structural composite
Transverse direction
Whisker
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616 • Chapter 16 / Composites
REFERENCES
Agarwal, B. D. and L. J. Broutman, Analysis and
Performance of Fiber Composites, 2nd edition,
Wiley, New York, 1990.
Ashbee, K. H., Fundamental Principles of Fiber Reinforced Composites, 2nd edition, Technomic
Publishing Company, Lancaster, PA, 1993.
ASM Handbook, Vol. 21, Composites, ASM International, Materials Park, OH, 2001.
Chawla, K. K., Composite Materials Science and
Engineering, 2nd edition, Springer-Verlag, New
York, 1998.
Chou, T. W., R. L. McCullough, and R. B. Pipes,
“Composites,” Scientific American, Vol. 255,
No. 4, October 1986, pp. 192–203.
Hollaway, L. (Editor), Handbook of Polymer Composites for Engineers, Technomic Publishing
Company, Lancaster, PA, 1994.
Hull, D. and T. W. Clyne, An Introduction to Composite Materials, 2nd edition, Cambridge University Press, New York, 1996.
Mallick, P. K., Fiber-Reinforced Composites, Materials, Manufacturing, and Design, 2nd edition,
Marcel Dekker, New York, 1993.
Peters, S. T., Handbook of Composites, 2nd edition,
Springer-Verlag, New York, 1998.
Strong, A. B., Fundamentals of Composites: Materials, Methods, and Applications, Society of
Manufacturing Engineers, Dearborn, MI, 1989.
Woishnis, W. A. (Editor), Engineering Plastics and
Composites, 2nd edition, ASM International,
Materials Park, OH, 1993.
QUESTIONS AND PROBLEMS
Large-Particle Composites
16.1 The mechanical properties of cobalt may be
improved by incorporating fine particles of
tungsten carbide (WC). Given that the moduli of elasticity of these materials are, respectively, 200 GPa (30 ⫻ 106 psi) and 700 GPa
(102 ⫻ 106 psi), plot modulus of elasticity
versus the volume percent of WC in Co from
0 to 100 vol%, using both upper- and lowerbound expressions.
16.2 Estimate the maximum and minimum thermal conductivity values for a cermet that
contains 90 vol% titanium carbide (TiC) particles in a nickel matrix. Assume thermal
conductivities of 27 and 67 W/m-K for TiC
and Ni, respectively.
16.3 A large-particle composite consisting of tungsten particles within a copper matrix is to be
prepared. If the volume fractions of tungsten
and copper are 0.70 and 0.30, respectively, estimate the upper limit for the specific stiffness
of this composite given the data that follow.
Copper
Tungsten
Specific
Gravity
Modulus of
Elasticity (GPa)
8.9
19.3
110
407
16.4 (a) What is the distinction between cement
and concrete?
(b) Cite three important limitations that
restrict the use of concrete as a structural
material.
(c) Briefly explain three techniques that
are utilized to strengthen concrete by reinforcement.
Dispersion-Strengthened Composites
16.5 Cite one similarity and two differences between precipitation hardening and dispersion strengthening.
Influence of Fiber Length
16.6 For some glass fiber–epoxy matrix combination, the critical fiber length–fiber diameter
ratio is 40. Using the data in Table 16.4, determine the fiber–matrix bond strength.
16.7 (a) For a fiber-reinforced composite, the efficiency of reinforcement h is dependent on
fiber length l according to
h⫽
l ⫺ 2x
l
where x represents the length of the fiber at
each end that does not contribute to the load
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Questions and Problems • 617
transfer. Make a plot of h versus l to l ⫽ 50
mm (2.0 in.) assuming that x ⫽ 1.25 mm
(0.05 in.).
(b) What length is required for a 0.90 efficiency of reinforcement?
Influence of Fiber Orientation and Concentration
16.8 A continuous and aligned fiber-reinforced
composite is to be produced consisting of
45 vol% aramid fibers and 55 vol% of a polycarbonate matrix; mechanical characteristics
of these two materials are as follows:
Aramid fiber
Polycarbonate
Modulus
of Elasticity
[GPa (psi)]
Tensile
Strength
[MPa (psi)]
131 (19 ⫻ 106)
2.4 (3.5 ⫻ 105)
3600 (520,000)
65 (9425)
Also, the stress on the polycarbonate matrix
when the aramid fibers fail is 35 MPa
(5075 psi).
For this composite, compute
(a) the longitudinal tensile strength, and
(b) the longitudinal modulus of elasticity
16.9 Is it possible to produce a continuous and
oriented aramid fiber–epoxy matrix composite having longitudinal and transverse
moduli of elasticity of 35 GPa (5 ⫻ 106 psi)
and 5.17 GPa (7.5 ⫻ 105 psi), respectively?
Why or why not? Assume that the elastic
modulus of the epoxy is 3.4 GPa 14.93 ⫻ 105
psi).
16.10 For a continuous and oriented fiberreinforced composite, the moduli of elasticity
in the longitudinal and transverse directions
are 33.1 and 3.66 GPa (4.8 ⫻ 106 and
5.3 ⫻ 105 psi), respectively. If the volume
fraction of fibers is 0.30, determine the moduli of elasticity of fiber and matrix phases.
16.11 (a) Verify that Equation 16.11, the expression for the fiber load–matrix load ratio
(Ff ⲐFm), is valid.
(b) What is the Ff ⲐFc ratio in terms of Ef, Em,
and Vf?
16.12 In an aligned and continuous carbon fiberreinforced nylon 6,6 composite, the fibers are
to carry 97% of a load applied in the longitudinal direction.
(a) Using the data provided, determine
the volume fraction of fibers that will be
required.
(b) What will be the tensile strength of this
composite? Assume that the matrix stress at
fiber failure is 50 MPa (7250 psi).
Carbon fiber
Nylon 6,6
Modulus
of Elasticity
[GPa (psi)]
Tensile
Strength
[MPa (psi)]
260 (37 ⫻ 106)
2.8 (4.0 ⫻ 105)
4000 (580,000)
76 (11,000)
16.13 Assume that the composite described in
Problem 16.8 has a cross-sectional area of
480 mm2 (0.75 in.2) and is subjected to a longitudinal load of 53,400 N (12,000 lbf).
(a) Calculate the fiber–matrix load ratio.
(b) Calculate the actual loads carried by
both fiber and matrix phases.
(c) Compute the magnitude of the stress on
each of the fiber and matrix phases.
(d) What strain is experienced by the composite?
16.14 A continuous and aligned fibrous reinforced
composite having a cross-sectional area of
970 mm2 (1.5 in.2) is subjected to an external tensile load. If the stresses sustained by
the fiber and matrix phases are 215 MPa
(31,300 psi) and 5.38 MPa (780 psi), respectively, the force sustained by the fiber phase
is 76,800 N (17,265 lbf), and the total longitudinal composite strain is 1.56 ⫻ 10⫺3, then
determine
(a) the force sustained by the matrix phase
(b) the modulus of elasticity of the composite material in the longitudinal direction, and
(c) the moduli of elasticity for fiber and matrix phases.
16.15 Compute the longitudinal strength of an
aligned carbon fiber–epoxy matrix composite having a 0.20 volume fraction of fibers,
assuming the following: (1) an average fiber
diameter of 6 ⫻ 10⫺3 mm (2.4 ⫻ 10⫺4 in.),
(2) an average fiber length of 8.0 mm (0.31 in.),
(3) a fiber fracture strength of 4.5 GPa
(6.5 ⫻ 105 psi), (4) a fiber–matrix bond
strength of 75 MPa (10,900 psi), (5) a matrix
stress at composite failure of 6.0 MPa (870 psi),